Abstract:

The Collatz problem is one of the most famous and notoriously hard problems in mathematics. Recall that the Collatz map sends an even number n to n/2 and an odd n to 3n+1. The Collatz conjecture states that starting from any natural number, after a finite number of iterations of the Collatz map, we get the value 1.
In 2008, Hicks, Mullen, Yucas and Zavislak suggested looking at a variant of the Collatz problem, where one considers polynomials in F_2[x], and the map sends a polynomial p such that p(0)=0 to p/x, and otherwise sends p to (x+1)p+1. In other words, we perform the Classical Collatz map in base-2 representation, without carrying the digits in addition and multiplication. They proved that starting from any nonzero polynomial p, we get to the polynomial 1 in at most O((deg p)^2) steps.
In a recent work with Angelot Behajaina and Elad Paran, we improved the above bound to O((deg p)^1.5). We will outline the proof and discuss related results and conjectures.